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ID: INPUT{id.yaml}

Title: >
  $abc$-triples of high quality

Definition: >
  The list contains all known $abc$-triples quality at least $1.4$,
  as givin in CITE{ABCatHomeHighQ}.
  An $abc$-triple is a solution of the equation $a+b=c$ with coprime integers
  $0 < a \leq b < c$.
  The quality of an $abc$-triple is defined as
  $q = \log(c)/\log\text{rad}(abc)$, 
  where $\text{rad}(abc)$ is the radical of $abc$, that is, 
  the product over all prime divisors of $abc$.
  
Parameters:
  q:
    type: R
    constraints: $q > 1.4$
  v:
    title: variable
    type: Symbolic
    show-in-parameter-list: no
   
Comments:
  comment-abc-conjecture: >
    The abc-conjecture CITE{WikiABC} states that $\limsup q = 1$, that is,
    for every $\varepsilon > 0$ there are only finitely many $abc$-triples
    of quality $q > 1+\varepsilon$.
    
Formulas:

Programs:

References:

Links:
  ABCatHome:
    title: "Bart de Smit: ABC@home" 
    url: https://www.math.leidenuniv.nl/~desmit/abc/
  ABCatHomeHighQ:
    title: "Bart de Smit: ABC@home: High quality triples" 
    url:     https://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2
  WikiABC:
    title: "Wikipedia: abc conjecture"
    url: https://en.wikipedia.org/wiki/Abc_conjecture
    
Similar tables:
  
Keywords:
  
Tags:
- number theory
- abc conjecture

Data properties:
  type: Z
  complete: unknown
  sources:
  - CITE{ABCatHomeHighQ} and the references therein
  reliability: >
    The quality $q$ is correct up to one ulp (unit of least precision); 
    see HREF{https://www.numberdb.org/help#section-number-types}.
Display properties:
  number-header: value

Numbers: INPUT{numbers.yaml}

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$abc$-triples of high quality

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Numbers

$q$

value

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Definition

The list contains all known $abc$-triples quality at least $1.4$, as givin in [2]. An $abc$-triple is a solution of the equation $a+b=c$ with coprime integers $0 < a \leq b < c$. The quality of an $abc$-triple is defined as $q = \log(c)/\log\text{rad}(abc)$, where $\text{rad}(abc)$ is the radical of $abc$, that is, the product over all prime divisors of $abc$.

Parameters

$q$

— real number ($q > 1.4$)

Comments

(1)

The abc-conjecture [3] states that $\limsup q = 1$, that is, for every $\varepsilon > 0$ there are only finitely many $abc$-triples of quality $q > 1+\varepsilon$.

Links

[1]

Bart de Smit: ABC@home

[2]

Bart de Smit: ABC@home: High quality triples

[3]

Wikipedia: abc conjecture

Data properties

Entries are of type: integer

Table is complete: unknown

Sources of data: [2] and the references therein

Reliability: The quality $q$ is correct up to one ulp (unit of least precision); see https://www.numberdb.org/help#section-number-types.